Supersymmetric QCD corrections to $e^+e^-\to t\bar{b}H^-$ and the Bernstein-Tkachov method of loop integration

0.23 MB·English

by  B. A. Kniehl

DESY 10-153
HD-THEP-10-17
MPP-2010-126
Supersymmetric QCD corrections to e+e− → t¯bH− and the
Bernstein-Tkachov method of loop integration
B. A. Kniehl1, M. Maniatis2 and M. M. Weber3
1II. Institut fu¨r Theoretische Physik, Universit¨at Hamburg,
Luruper Chaussee 149, 22761 Hamburg, Germany
2Institut fu¨r Theoretische Physik, Universit¨at Heidelberg,
1
Philosophenweg 16, 69120 Heidelberg, Germany
1
0
3Max-Planck-Institut fu¨r Physik (Werner-Heisenberg-Institut),
2
Fo¨hringer Ring 6, 80805 Mu¨nchen, Germany
n
a
J
Abstract

1
1 ThediscoveryofchargedHiggsbosonsisofparticularimportance,sincetheirex-

istence is predicted by supersymmetry and they are absent in the Standard Model
]
h (SM). If the charged Higgs bosons are too heavy to be produced in pairs at fu-
p
ture linear colliders, single production associated with a top and a bottom quark is
-
p enhancedin partsof theparameter space. We presentthenext-to-leading-order cal-
e
h culation in supersymmetric QCD within the minimal supersymmetricSM (MSSM),
[ completing a previous calculation of the SM-QCD corrections. In addition to the

2 usual approach to perform the loop integration analytically, we apply a numerical
v approach based on the Bernstein-Tkachov theorem. In this framework, we avoid
9
some of the generic problems connected with the analytical method.
2
9
3
.
9
0
0
1
:
v
i
X
r
a
1 Introduction
±
The discovery of charged Higgs bosons (H ) would beinstant evidence for physics beyond
the SM, which only accommodates a single neutral Higgs boson. Charged Higgs bosons
appear in models with two Higgs doublets, as are required by supersymmetric (SUSY)
extensions of the SM. For this reason, there is much interest in charged-Higgs-boson
physics (review articles on this subject include, for instance, Refs. [1, 2, 3]).
Phenomenologically, theLargeHadronCollider(LHC)willbethefirstcolliderwiththe
± ±
potential to discover theH bosons. If the charged-Higgs-bosonmass m is nottoolarge,
H
±
i.e. if m < m m , the dominant production channel is via top-quark pair production
H t − b
gg tt¯with subsequent decay of a top quark or antiquark into a charged Higgs boson,
t →bH+ or t¯ ¯bH−, respectively. If the H± bosons are too heavy for this subsequent
→ →
top-quark decay, then the dominant production channel would be bottom-gluon fusion,
gb tH− and g¯b t¯H+ [4, 5, 6, 7, 8]. In all these processes, the charged-Higgs-boson
→ →
signal has to be carefully separated from large SM-QCD background at hadron colliders.
Despite the fact that charged Higgs bosons may be discovered at the LHC, a precise
determination of their properties will only be possible at a linear collider, such as the
proposed international linear collider (ILC). If the value of mH± is not too large, i.e. if
mH± < √s/2, the dominant production channel will be e+e− H+H− [9, 10]. On the
→
other hand, this production channel may not be accessible at the ILC because of the
limited center-of-mass energy √s. In this case, charged Higgs bosons may be copiously
produced singly via the two channels [3]
e+e− τ+ν H− +c.c. (1)
τ
→
e+e− t¯bH− +c.c. . (2)
→
The first production channel (1) was investigated in Ref. [3], where single charged-Higgs-
boson production processes were systematically compared with each other with leading-
order (LO) accuracy. Here, we focus on the second production channel (2), which is
enhanced in parts of the parameter space. Since QCD corrections are typically large, we
present a computation with next-to-leading-order (NLO) accuracy, to (α ).
s
O
One part of the NLO calculation consists of the SM-QCD contribution, i.e. the purely
gluonic corrections, which were already presented in Ref. [11]. Here, we add the SUSY-
QCDcontribution, duetosquark andgluinoloops, inthe minimal SUSYSM(MSSM) [12,
13, 14]. Since the MSSM makes no assumption about the SUSY-breaking mechanism,
but just uses explicit SUSY-breaking terms in the Lagrangian, it may be considered as
representative for a wide class of SUSY models. The SM-QCD and SUSY-QCD parts
taken together yield a complete prediction with (α ) accuracy.
s
O
In the calculation of the virtual corrections, we encounter loop integrals. Using the
conventional analytic approach of Ref. [15], all scalar loop integrals can be expressed
in terms of logarithms and dilogarithms. Furthermore, using the reduction algorithm of
Ref.[16], alltensorloopintegrals, i.e.integralscontainingloopmomentainthenumerator,
can be expressed in terms of scalar integrals. Therefore, a full analytic solution for one-
loopintegrals exists. However, in general, this approach hasa number of drawbacks. First
of all, the number of dilogarithms in the analytic expression of a scalar integral increases
rapidly with an increasing number of external legs. This may lead to cancellations for
1
t t t
¯b
H− ¯b
γ,Z γ,Z γ,Z
¯b
H− H−
Figure 1: Feynman diagrams of the process e+e− t¯bH− at Born level. Those of the
charge-conjugated process e+e− bt¯H+ are not sho→wn.
→
multileg integrals in certain kinematic regions [17]. Furthermore the tensor reduction
of Ref. [16] introduces inverse Gram determinants. These may vanish at the phase-
space boundary, even though the tensor coefficients themselves remain regular there.
There may thus be cancellations among terms in the numerator, which may lead to
numerical instabilities. While these generic problems may still be overcome in the case
under consideration here, where at most box diagrams occur, by exercising care in the
phase-space integrations, they become quite severe for five and more external legs. To
address these problems, several improved reduction algorithms have been constructed [18,
19, 20, 21] allowing a numerically stable evaluation of the tensor integral coefficients. A
different solution, typically used in the context of nondiagrammatic methods to calculate
loop amplitudes, is to employ high-precision arithmetics for potentially unstable phase-
space points (see Refs. [22, 23, 24] for reviews of these techniques). However, these
improvements come at the price of either increased complexity of the more elaborate
reduction algorithms or increased runtime from the high-precision evaluations.
Finally, within dimensional regularization in D = 4 2ǫ space-time dimensions, the
−
evaluation of the loop-by-loop contribution to a two-loop correction makes it necessary to
expand the one-loop integrals beyond the constant term in the expansion about D = 4
dimensions. An analytic calculation of these higher-order terms is rather complicated.
Weexplore hereanalternative numerical approachtotheevaluationofone-looptensor
and scalar integrals. This strategy is described in detail in Ref. [25] and is based on the
Bernstein-Tkachov (BT) theorem [26], which can be used to rewrite one-loop integrals in
Feynman-parametric representation in a formbetter suited fornumerical evaluation. This
technique yields a fast and reliable numerical calculation of multileg one-loop integrals.
This method is numerically stable also for exceptional momentum configurations and
easily allows for the introduction of complex masses and the calculation of higher orders
of the expansion about D = 4 dimensions. Numerical methods based on the BT theorem
have also been developed for two-loop self-energy and vertex integrals [27, 28, 29, 30], and
found several applications [31, 32, 33, 34]. While these applications involved two-loop
integrals with full mass dependence, the phase-space integrations were trivial. We adopt
the BT method in our calculation and compare it to the conventional analytic approach
in order to investigate its performance in the computation of cross sections including
nontrivial phase-space integrations at one loop.
2
(a) (b) (c) (d) (e)
t t
b t g˜ t˜x b
t
˜by b t˜x H g˜ ˜by g˜
g˜ H− g˜ b ˜by H− ˜by b t˜x t
˜bx t ˜bx b H− ˜by
b t˜y H− H−
Figure 2: Representative one-loop diagrams of the SUSY-QCD corrections to the process
e+e− t¯bH−. The inner lines represent gluinos (g˜) as well as the various top (t˜x) and
bottom→(˜bx) squarks with x = 1,2 labeling the mass eigenstates.
2 The calculation
We consider the process e+e− t¯bH− and its charge-conjugate counter part e+e−
→ →
bt¯H+. The corresponding LO Feynman diagrams are shown in Fig. 1. Since the cross
sections for both processes are identical due to CP invariance, we only show results for
¯ −
the tbH final state in the following. Furthermore, we only consider the kinematical
regime where neither the intermediate Higgs bosons nor the top quark can be on shell
and, therefore, we do not have to include the respective finite widths.
The SUSY-QCD corrections arise from loop contributions containing gluinos, stops,
and sbottoms in the propagators; see Fig. 2 for some representative diagrams. This
virtual contribution may be classified into two-, three-, and four-point integrals. Since
the particles in the loops are massive, no infrared or collinear singularities arise and no
real-emission contribution has to be included.
The calculation is performed with the help of the program packages FeynArts [35]
and FormCalc [36]. The analytical expressions from FormCalc are post-processed and
translated to a C++ code. The one-loop integrals are evaluated using the BT method as
described in detail below. We perform a second calculation using analytical loop integrals
as implemented by LoopTools [36]. This calculation is also based on FeynArts and
FormCalc. The phase-space integration is performed with the Monte Carlo integration
routine Vegas [37] in both cases.
The renormalization of the strong-coupling constant α is performed in the modified
s
minimal-subtraction (MS) scheme of dimensional regularization with the squarks and
gluinos decoupled from the running. The top quark only contributes to the running of α
s
abovethescalem , byincreasingthenumberofactivequarkflavorsfromn = 5ton = 6.
t f f
The quark masses and wave functions are renormalized on shell, including the top-quark
mass in the Yukawa coupling. For the bottom-quark mass in the Yukawa coupling, we
use the running QCD MS version m (µ), with µ being the renormalization scale, and
b
optionally perform a resummation of large SUSY corrections as described below. We use
the running bottom-quark mass, since the pure QCD corrections contain large logarithms
of the type log(µ/m ) originating from the Yukawa interaction. These can be resummed
b
by using the running bottom-quark mass m (µ) [38, 39]. The calculation of the purely
b
gluonic QCD corrections showed that the bulk of the QCD corrections may be absorbed
by using the running bottom-quark mass [11]. Since the squark and gluino masses enter
3
only at NLO we do not need to renormalize them.
In the MSSM, two Higgs-boson doublets, denoted by Hˆ = (H+,H0) and Hˆ =
u u u d
(H0,H−), are required. The H0 field is responsible for the generation of the up-type-
d d u
fermion masses and the H0 field for the down-type-fermion masses, i.e. the bottom quark
d
couples to H0 but not to H0. Nevertheless, the coupling of the bottom quark to the H0
d u u
field is dynamically generated via loops [40]. Although this coupling is loop suppressed,
once the H0 fields acquire their vacuum expectation values v , a large value of v
u/d u/d u
may compensate a small loop contribution, i.e. these effects may be considerable for large
values of tanβ = v /v . These large tanβ-enhanced contributions may be resummed to
u d
all orders [40] by replacing the bottom-quark mass in the Yukawa coupling as
m (µ) ∆m
b b
m 1 , (3)
b → 1+∆m − tan2β
b (cid:18) (cid:19)
where
2α (µ)
s
∆m = m µ˜tanβ I(m ,m ,m ),
b 3π g˜ g˜ ˜b1 ˜b2
1 a2 b2 c2
I(a,b,c) = a2b2log +b2c2log +c2a2log . (4)
(a2 b2)(b2 c2)(a2 c2) b2 c2 a2
− − − (cid:18) (cid:19)
Here, µ˜ is the Higgs-Higgsino mass parameter of the superpotential. In order to prevent
double counting, an extra counterterm of the form
1
δmYuk = m (µ)∆m 1+ (5)
b b b tan2β
(cid:18) (cid:19)
for the Yukawa coupling is needed at NLO. The resummation formalism was extended to
alsoinclude thedominant termsinthetrilinear coupling A [41]. Since thesecontributions
b
are small for our parameter values, we do not include them in the resummation.
2.1 Numerical evaluation of loop integrals
Within dimensional regularization, any scalar one-loop integral can be expressed as an
integral over Feynman parameters as
(2πµ)4−D 1
ID = dDq
N iπ2 Z [q2 −m21][(q +p1)2 −m22]···[(q +pN−1)2 −m2N] (6)
= (4πµ2)ǫ Γ(N 2+ǫ)( 1)N dSN−1V(xi)−(N−2+ǫ),
− −
Z
where
1 x1 xn−1
dS = dx dx dx
n 1 2 n
···
Z Z0 Z0 Z0
and V is a quadratic form in the N 1 Feynman parameters x ,
i
−
V(x) = xTHx+2KTx+L iδ.
−
4
The coefficients H, K, and L of V are given in terms of the momenta p and the masses
i
m .
i
In general, the quadratic form V can vanish within the integration region, although,
strictly speaking, the zeros are shifted into the complex plane by the infinitesimal imagi-
nary part iδ. Since the limit δ 0 has to be taken in the end, the form given above is in
→
general not suited for direct numerical integration.
Instead, the integral can be rewritten using the BT theorem [26] before attempting a
numerical evaluation. Applied to the case of one-loop integrals, this theorem states that,
for any quadratic form V(x) raised to any real power β, we have
(x X) ∂
1 − i i V1+β(x ) = B Vβ(x ), (7)
i i
− 2(1+β) ·
(cid:20) (cid:21)
where X = KTH−1, B = L KTH−1K, and ∂ = ∂/∂x . Inserting this relation into a
i i
− −
Feynman-parameter integral and integrating by parts, one obtains
n
1
dSnVβ = 2B(1+β) (2+n+2β) dSnV1+β − dSn−1 χiVi1+β , (8)
" #
Z Z Z i=0
X
where χ = X X , with X = 1 and X = 0, and
i i i+1 0 n+1
−
V(1,x1,...,xn−1) for i = 0,
Vi(x1,...,xn−1) = V(x1,...,xi,xi,...,xn−1) for 0 < i < n,
V(x1,...,xn−1,0) for i = n.

Applied to the one-loop integral of Eq. (6), the first term inside the brackets in Eq. (8)
corresponds to the N-point integral in D + 2 dimensions, while the last term is a sum
over (N 1)-point integrals in D dimensions obtained by removing one propagator.
−
Recursive application of Eq. (8) allows us to express any scalar one-loop integral as a
linear combination of terms of the form dS V(x )m−ǫ with any integer m 0. A Taylor
k i
≥
expansion up to (ǫa) then results in terms of the form dS Vm log1+aV. For m = 0,
k
O R ·
the integrand still contains an integrable (logarithmic) singularity, while it is smooth for
R
m > 0. Although larger values of m lead to smoother integrands, the expressions also
grow larger due to the repeated application of the BT identity (8). The optimal choice
for m depends on the chosen numerical integration routine and its ability to deal with
integrable singularities.
The parametric representation of tensor integrals contains Feynman parameters in the
numerator. The procedure outlined above can also be applied in this case, so that no
separate reduction to scalar integrals is needed. Furthermore, no inverse Gram determi-
nants are introduced using this approach, making the latter numerically reliable also for
exceptional kinematic configurations.
The BT identity (8) still contains a potentially small factor of B in the denominator.
Although the zeros of B correspond to the leading Landau singularities, the singular
behavior in the vicinity of this singularity is overestimated by the factor 1/B. This may
result in numerical cancellations in the numerator leading to instabilities. We, therefore,
5
use an alternative BT-like relation for the three-point function [42], namely
(x X) ∂ n=∞ ( ǫ)n−1 1 Q(x)
V−1−ǫ(x) = B−ǫ 1+ − i i − lnn 1+ ,
2 n! Q(x) B
(cid:20) (cid:21) n=1 (cid:18) (cid:19)
X
where Q(x) is defined by the decomposition of V(x) = Q(x)+B. For small values of B,
the 1/B behavior is reduced to lnB, which is in agreement with the singular behavior
near the leading Landau singularity of a triangle diagram.
We use an implementation of the BT method in Mathematica and C++ [43] that
allows for the calculation of all the appearing one-loop tensor coefficients. The Feynman-
parameter integration is performed with a deterministic integration routine, with the
number of integrand evaluations limited to 105. We verify the results for the virtual
corrections using the BT approach against an analytic evaluation of the integrals with
LoopTools [36] for single points in phase-space and find good agreement within numerical
integration errors.
We could now perform the phase-space integration of the virtual corrections using
Monte Carlo techniques. However, using the BT approach for the evaluation of the loop
integrals requires a separate numerical integration for each integral at every phase-space
point. This turns out to be very inefficient compared to the analytic evaluation of the
loop integrals.
Therefore, we combine the phase-space integration and the integration over the Feyn-
manparametersintoasingleintegrationthatisperformedusingtheadaptiveMonteCarlo
integration program Vegas [37]. The adaptivity of Vegas optimizes the phase-space and
Feynman-parameter integrations at the same time, leading to a significant improvement
of the efficiency. All results for the BT approach shown below are obtained using this
combined integration.
3 Results
Before we present numerical results, we fix the input parameters. We use the following
SM parameters:
α = 1/137.0359998, α (m ) = 0.1184,
s Z
m = 80.398GeV, m = 91.1876GeV, (9)
W Z
m = 173.1GeV, m (m ) = 4.2GeV.
t b b
The running MS bottom-quark mass m (m ) is used as input and corresponds to an on-
b b
shell mass of mos = 4.58GeV. As mentioned above, we use the on-shell bottom-quark
b
±
mass everywhere, except for the tbH Yukawa coupling, for which we always use the
running MS version. Both m (µ) and α (µ) are evaluated at the renormalization scale
b s
µ = (mH± +mt+mb)/3. The relevant scale for the strong coupling appearing in ∆mb in
Eq. (4) is µ = (m +m +m )/3. However, the two-loop result of Refs. [44, 45, 46] for
b g˜ ˜b1 ˜b2
∆m develops a maximum at about µ /3 to µ /4, which is, therefore, a more appropriate
b b b
scale choice in ∆m . For our choice of Higgs-boson and squark masses, µ accidentally
b
falls into this region. For simplicity, we therefore use µ = µ in our calculation. We note
b
6
m0 m1/2 A0 tanβ sign µ mH± mg˜ m˜b1 m˜b2 mt˜1 mt˜2
SPS1a 100 250 100 10 + 404.5 607.7 514.4 543.8 400.7 586.8
−
SPS4 400 300 0 50 + 343.3 734.4 617.1 682.5 548.8 698.7
Table 1: Definition of the Snowmass points SPS1a and SPS4. The values of the masses
and the trilinear coupling A are given in units of GeV. The left part of the table shows the
0
mSUGRA parameters the SPS points correspond to (the scalar and gaugino mass param-
eters m and m , the trilinear coupling A , the ratio of the Higgs vacuum expectation
0 1/2 0
values tanβ, and the sign of the SUSY Higgs-boson mass parameter µ). The right part
shows the masses obtained from Softsusy via renormalization group running from the
high-energy scale to the weak scale (the charged-Higgs-boson mass mH±, the gluino mass
m , and the sbottom and stop masses m , m , m , m ).
g˜ ˜b1 ˜b2 t˜1 t˜2
that a shift in the renormalization scale of α (µ) in Eq. (4) formally creates a contribution
s
beyond the order of our calculation.
In order to fix the MSSM parameters, we adopt the scenarios Snowmass Points and
Slopes (SPS) 1a and SPS4 [47, 48], which are both derived from minimal supergravity
(mSUGRA). While SPS1a is a typical mSUGRA scenario with a moderate value of tanβ,
tanβ = 10, the SPS4 point has a high value, tanβ = 50. In the SPS4 scenario, we there-
foreexpect largecorrections fromtanβ-enhancedcontributions. The input parametersfor
both scenarios are summarized in Table 1. They are defined at a high-energy unification
scale. We use the program Softsusy2.0.17 [49] to perform the renormalization group
evolution from the high-energy scale to the weak scale and to calculate the spectrum from
theSPSinput parameters. Theresultingcharged-Higgs-bosonandrelevant SUSY-particle
masses are also shown in Table 1. The latter are defined in Softsusy2.0.17 according to
the modified minimal-subtraction (DR) scheme of dimensional reduction. They enter our
core calculations only at one loop, so that a change of scheme would induce a shift beyond
NLO. For the transition to the MS scheme, such a shift would be numerically insignificant
[50]. In order to study the dependence on tanβ and mH±, we vary the high-scale input
parameters and recalculate the full spectrum. While tanβ can be directly used as an
input parameter to the spectrum calculation, we adjust m to obtain the desired value of
0
mH±. This also results in a change of the squark masses.
We are interested in a kinematical region where the center-of-mass energy √s is not
sufficient to produce pairs of charged Higgs bosons, i.e. √s < 2mH±, but sufficient to
produce all final-state particles on-shell, i.e. √s > mH±+mb+mt. For a 800 GeV collider,
this implies that the interesting region lies in the range 400GeV < mH± < 620GeV.
In Fig. 3, the total cross section σ(e+e− t¯bH−) is shown for the SPS1a scenario as a
→
function ofmH± intherange√s/2 < mH± < √s mb mt. The cross section fallsrapidly
− −
with increasing distance from the pair-production threshold at mH± = √s/2 = 400GeV.
Before resummation, the SUSY-QCD corrections range from 22% to 24% and exhibit
− −
onlyaweakdependence onmH±. Afterresummation, theresidual correctionsstillamount
to about 15%, which implies that the tanβ-enhanced contributions are not actually
−
dominant for these values of parameters.
Figure 4 shows the mH± dependence for the SPS4 scenario with tanβ = 50. In this
case, theSUSY-QCD correctionsaremuch larger, ranging from 40%to 55%. However,
− −
7
σ[fb] δ[%]
SPS1a,√s=800GeV SPS1a,√s=800GeV
0.01 0
LOnonres nonres
NLOnonres resum
LOresum 5
0.001 NLOresum −
10
−
1 10−4
×
15
−
1 10−5
×
20
−
1 10−6
× 25
−
1 10−7 30
× 400 420 440 460 480 500 520 540 560 580 600 − 400 420 440 460 480 500 520 540 560 580 600
mH[GeV] mH[GeV]
Figure 3: Total cross section σ(e+e− t¯bH−) (left panel) and relative corrections δ in
→
percent (right panel) depending on the charged-Higgs-boson mass mH± for a center-of-
mass energy of √s = 800 GeV. The other MSSM parameters are taken to have their
SPS1a values. Both the nonresummed and resummed cross sections are considered.
σ[fb] δ[%]
SPS4,√s=800GeV SPS4,√s=800GeV
0.1 0
LOnonres nonres
NLOnonres resum
LOresum 10
0.01 NLOresum −
20
−
0.001
30
−
1 10−4
×
40
−
1 10−5
× 50
−
1 10−6 60
× 400 420 440 460 480 500 520 540 560 580 600 − 400 420 440 460 480 500 520 540 560 580 600
mH[GeV] mH[GeV]
Figure 4: Total cross section σ(e+e− t¯bH−) (left panel) and relative corrections δ in
→
percent (right panel) depending on the charged Higgs-boson mass mH± for a center-of-
mass energy of √s = 800 GeV. The other MSSM parameters are taken to have their
SPS4 values. Both the nonresummed and resummed cross sections are considered.
8
σ[fb] δ[%]
SPS1a,√s=800GeV SPS1a,√s=800GeV
0.07 0
LOnonres nonres
NLOnonres resum
0.06 LOresum
NLO resum 10
−
0.05
20
−
0.04
30
−
0.03
40
0.02 −
50
0.01 −
0 60
10 15 20 25 30 35 40 45 50 − 10 15 20 25 30 35 40 45 50
tanβ tanβ
Figure 5: Total cross section σ(e+e− t¯bH−) (left panel) and relative corrections δ in
→
percent (right panel) as a function of tanβ for a center-of-mass energy of √s = 800 GeV.
The other MSSM parameters have their SPS1a values. Both the nonresummed and re-
summed cross sections are considered.
the bulk of these corrections is absorbed by the resummation. The size of the residual
corrections ranges from 8% to 17% and is, therefore, similar to the SPS1a scenario.
In Fig. 5, the tanβ d−ependen−ce of the total cross section σ(e+e− t¯bH−) is shown
→
for the SPS1a scenario. The cross section grows strongly with increasing value of tanβ.
Without resummation, the relative corrections also show a strong tanβ dependence and
range from 24% at low value of tanβ up to 55% at high value of tanβ. Most of these
− −
corrections are absorbed by resummation; the remaining corrections range from 10% to
−
15% and show only a relatively weak dependence on tanβ. Since we change tanβ by
−
modifying the high-energy input parameters, we also need to change m accordingly to
0
keep the low-energy value of mH± at its desired value. This leads to somewhat higher
squark masses at high values of tanβ, so that the loop corrections are suppressed there.
Numerically, we find that the competing effects of the loop suppression due to heavier
squark masses and of the increase in the bottom Yukawa coupling lead to a maximum
in the relative corrections at tanβ 45. The difference between the nonresummed and
≈
resummed NLOcross sections measures thesize ofthehigher-order contributions included
byperformingtheresummation. Forhighvaluesoftanβ, thesecontributionsareofsimilar
size to the residual SUSY-QCD corrections underlining the need for resummation in this
region.
We note that the analogous analysis of the SUSY-QCD corrections to the associated
production of a neutral Higgs boson with a b¯b pair in e+e− annihilation with and without
resummation of tanβ-enhanced contributions yielded similar results [51].
9